It's not so much that you're not productive when depressed, as the awareness that achievement in other spheres of activity are not actually all that satisfying absent a nurturing relationship - also a problem for many lottery winners who find themselves trying and failing to consume happiness.
I'm sorry that happened but you shouldn't treat it as an insurmountable problem. If you're just 20, your body isn't even completely done growing and you're also relatively inexperienced in dealing with intense emotions. You'll always bear the scars of the pain you felt from this relationship breaking up, and have regrets about whatever parts of it were caused by you, but you can and will heal. Being willing to accept help through therapy and reassess yourself means you're already a long way towards that.
Be careful of the hesitancy to get into another relationship - a natural doubt, but one that can stiffen into a 'reaction-formation' is taken too far. It's certainly tempting (especially when you're younger) to just date casually for social and sexual companionship, but if you systematically avoid relationships then undertaking and sustaining new ones will be more difficult, and as you get older the frustrations and loneliness that can result will weigh increasingly heavy and result in bitterness.
It may help to think of relating to others as a skill to be developed and continuously improved rather than as an achievement to be unlocked (which is way overstressed in media and marketing). A pet animal can help a lot with that. The obligations involved impose a sort of emotional discipline on you, but the rewards are unbounded as animals are generous with their affections. If you end up with a dog it will also improve your social life - people are well-disposed to the owner of a happy dog, and it also makes small talk much easier.